Hermitian and Unitary Matrices

Hermitian Conjugation of a Matrix

Definition.

Hermitian conjugate of a rectangular complex matrix \(\ \boldsymbol{A}\,=\,[\,\alpha_{ij}]_{m\times n}\ \) is a matrix \(\ \boldsymbol{A}^+=\,[\,\alpha_{ij}^+\,]_{n\times m}\,,\ \,\) where \(\ \alpha_{ij}^+\,:\,=\,\alpha_{ji}^*\,,\ \) \(i=1,2,\dots,n,\ \ j=1,2,\dots,m\,.\)

Operation of Hermitian conjugation is thus a composition of matrix transposition \(\\\) and complex conjugation of its elements (the last two operations are commutative):

(1)\[\boldsymbol{A}^+\,:\,=\ (\boldsymbol{A}^T)^*\,=\ (\boldsymbol{A}^*)^T\,.\]

The name comes from French mathematician Charles Hermite (1822-1901).

In analogy to complex conjugation, we will use the notion “Hermitian conjugation” also for an operation whose result is Hermitian conjugate of a matrix.

Example.

\[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{ccc} 2-i & 3 & 1+i \\ 0 & -1+2\,i & 4\,i \end{array}\right]\in M_{2\times 3}(C)\,:\quad \boldsymbol{A}^+\,=\ \left[\begin{array}{cc} 2+i & 0 \\ 3 & -1-2\,i \\ 1-i & -4\,i \end{array}\right]\in M_{3\times 2}(C)\,.\end{split}\]

Properties of Hermitian conjugation.

1. Hermitian conjugate of sum of matrices \(\ \boldsymbol{A},\boldsymbol{B}\in M_{m\times n}(C)\ \) is equal to sum of their Hermitian conjugates:

   \[(\boldsymbol{A}+\boldsymbol{B})^+\,=\ \boldsymbol{A}^+\,+\ \boldsymbol{B}^+\,.\]

2. Multiplication of a matrix by scalar \(\,\alpha\in C\ \) multiplies its Hermitian conjugate by \(\,\alpha^*:\)

   \[(\alpha\boldsymbol{A})^+\,=\ \alpha^*\boldsymbol{A}^+\,,\qquad \alpha\in C\,,\ \ \boldsymbol{A}\in M_{m\times n}(C)\,.\]

3. Hermitian conjugate of product of matrices \(\ \boldsymbol{A}\in M_{m\times p}\ \) and \(\ \boldsymbol{B}\in M_{p\times n}\ \) is equal to product of Hermitian conjugates with reverse order of the factors:

   \[(\boldsymbol{A}\boldsymbol{B})^+\,=\ \boldsymbol{B}^+\boldsymbol{A}^+\,.\]

4. Double Hermitian conjugation returns the initial matrix:

   \[(\boldsymbol{A}^+)^+\,=\ \boldsymbol{A}\,,\qquad\boldsymbol{A}\in M_{m\times n}(C)\,.\]

Corollary.

Hermitian conjugation is an antilinear operation:

\[(\alpha_1\boldsymbol{A}_1+\alpha_2\boldsymbol{A}_2)^+\,=\ \alpha_1^*\,\boldsymbol{A}_1^+\,+\,\alpha_2^*\,\boldsymbol{A}_2^+\,,\quad \alpha_1,\alpha_2\in C\,,\ \ \boldsymbol{A}_1,\boldsymbol{A}_2\in M_{m\times n}(C)\,.\]

For a real matrix \(\,\boldsymbol{A}\in M_{m\times n}(R),\ \) Hermitian conjugate boils down to transpose: \(\,\boldsymbol{A}^+\,=\ \boldsymbol{A}^T\,.\)

Now, an inner product of vectors \(\ \ \boldsymbol{x}\,=\, \left[\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \dots \\ \alpha_n \end{array}\right] \ \ \ \) and \(\quad \boldsymbol{y}\,=\, \left[\begin{array}{c} \beta_1 \\ \beta_2 \\ \dots \\ \beta_n \end{array}\right]\) in the space \(\,C^n\ \) may be concisely written in a form of a matrix product:

(2)\[\begin{split}\langle \boldsymbol{x},\boldsymbol{y}\rangle\ \,=\ \, \sum_{i\,=\,1}^n\ \alpha_i^*\,\beta_i\ \,=\ \, [\,\alpha_1^*,\,\alpha_2^*,\,\dots,\,\alpha_n^*\,]\ \left[\begin{array}{c} \beta_1 \\ \beta_2 \\ \dots \\ \beta_n \end{array}\right]\ \,=\ \, \boldsymbol{x}^+\,\boldsymbol{y}\,.\end{split}\]

Theorem 5.

For a given matrix \(\,\boldsymbol{A}\in M_n(C),\ \) \(\ \boldsymbol{A}^+\ \) is the only matrix satisfying the condition

(3)\[\langle\,\boldsymbol{x},\boldsymbol{A}^+\boldsymbol{y}\,\rangle\ =\ \langle\,\boldsymbol{A}\boldsymbol{x},\boldsymbol{y}\,\rangle\qquad \text{for every}\ \ \boldsymbol{x},\boldsymbol{y}\in C^n\,.\]

Proof.

The property 3. of Hermitian conjugation and the formula (2) imply that

\[\langle\boldsymbol{x},\boldsymbol{A}^+\boldsymbol{y}\rangle\,=\, \boldsymbol{x}^+(\boldsymbol{A}^+\boldsymbol{y})\,=\, (\boldsymbol{x}^+\boldsymbol{A}^+)\ \boldsymbol{y}\,=\, (\boldsymbol{A}\boldsymbol{x})^+\boldsymbol{y}\,=\, \langle\boldsymbol{A}\boldsymbol{x},\boldsymbol{y}\rangle\,.\]

Hence, the matrix \(\,\boldsymbol{A}^+\ \) satisfies the condition (3). To show that this is the only matrix with such property \(\,\) denote \(\,\boldsymbol{A}=[\,\alpha_{ij}\,]_{n\times n}\ \) and \(\,\) assume that for certain matrix \(\,\boldsymbol{B}=[\,\beta_{ij}\,]_{n\times n}:\)

\[\langle\,\boldsymbol{x},\boldsymbol{B}\boldsymbol{y}\,\rangle\ =\ \langle\,\boldsymbol{A}\boldsymbol{x},\boldsymbol{y}\,\rangle\qquad \text{for every}\ \ \boldsymbol{x},\boldsymbol{y}\in C^n\,.\]

In particular, if \(\ \,\boldsymbol{x},\,\boldsymbol{y}\ \,\) are the canonical basis vectors \(\ \,\boldsymbol{e}_i,\,\boldsymbol{e}_j\ ,\,\) we obtain \(\,\) (\(\ i,j=1,2,\dots,n\)) :

\[\beta_{ij}\,=\ \boldsymbol{e}_i^+\,\boldsymbol{B}\,\boldsymbol{e}_j\,=\ \langle\,\boldsymbol{e}_i,\boldsymbol{B}\boldsymbol{e}_j\rangle\ =\ \langle\,\boldsymbol{A}\boldsymbol{e}_i,\boldsymbol{e}_j\,\rangle\ =\ \langle\,\boldsymbol{e}_j,\boldsymbol{A}\boldsymbol{e}_i\rangle^*\ =\ (\boldsymbol{e}_j^+\boldsymbol{A}\;\boldsymbol{e}_i)^*\,=\ \alpha_{ji}^*\,=\ \alpha_{ij}^+\,,\]

which gives the equality \(\ \boldsymbol{B}=\boldsymbol{A}^+\,.\)

Hence, the condition (3) may be treated as an equivalent definition for Hermitian conjugate \(\ \boldsymbol{A}^+\,\) of a square matrix \(\,\boldsymbol{A}.\ \) Further we will see that exactly in this way one defines Hermitian conjugation of a linear operator.

Theorem 6.

Determinant of Hermitian conjugate of a square complex matrix is equal to complex conjugate of its determinant:

\[\det\boldsymbol{A}^+\ =\ (\det\boldsymbol{A})^*\,,\qquad\boldsymbol{A}\in M_n(C)\,.\]

Proof. \(\,\) Let \(\,\boldsymbol{A}=[\,\alpha_{ij}\,]_{n\times n}\in M_n(C).\) By definition (1), we have

\[\det\boldsymbol{A}^+\,=\ \det\,(\boldsymbol{A}^*)^T\,=\ \det\boldsymbol{A}^*\,, \qquad\text{where}\quad\boldsymbol{A}^*=[\,\alpha_{ij}^*\,]_{n\times n}\,.\]

Now it is easy to see from permutation expansion of the determinant that the determinant of complex conjugate of a matrix is equal to complex conjugate of its determinant: \(\ \,\det\boldsymbol{A}^*\equiv\det[\,\alpha_{ij}^*\,]\ =\ (\det\boldsymbol{A})^*\,,\ \,\) which leads directly to the hypothesis.

Hermitian Matrices

Definition.

Matrix \(\,\boldsymbol{A}=[\,\alpha_{ij}\,]_{n\times n}\in M_n(C)\ \) is a Hermitian matrix \(\,\) if it is equal to its Hermitian conjugation:

(4)\[\boldsymbol{A}\,=\,\boldsymbol{A}^+\,,\qquad\text{that is}\quad \alpha_{ij}=\alpha_{ji}^*\,,\quad i,j=1,2,\dots,n.\]

Example of a Hermitian matrix:

\[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{ccc} 3 & 2-i & -4+3\,i \\ 2+i & -1 & -i \\ -4-3\,i & i & 5 \end{array}\right]\,.\end{split}\]

The properties given below state that certain quantity related with a (complex) \(\,\) Hermitian matrix is real. To show that a complex number is a real number, it is useful to have the following

Lemma. \(\,\) Let \(\,z\in C.\ \,\) Then \(\quad z\in R\quad\Leftrightarrow\quad z=z^*\,.\)

Indeed, \(\,\) if \(\ z=a+b\,i\,,\ \) then the condition \(\ \,z=z^*\ \,\) means that \(\ \,a+b\,i=a-b\,i\,,\ \,\) \(\\\) which is equivalent to saying that \(\ \,b\equiv\text{im}\,z=0.\)

Properties of Hermitian matrix.

1. Diagonal entries of a Hermitian matrix are real numbers. \(\\\) Indeed, if we write the condition (4) for \(\,i=j\ \) we obtain \(\ \alpha_{ii}=\alpha_{ii}^*\,,\ \) \(\\\) which means that \(\ \alpha_{ii}\in R\,,\ \ i=1,2,\dots,n\,.\)

2. Trace and determinant of a Hermitian matrix are real: \(\ \text{tr}\,\boldsymbol{A},\,\det\boldsymbol{A}\,\in\,R\,.\) This follows from the definition of trace as a sum of diagonal entries of the matrix and from Theorem 6. about determinant of Hermitian conjugate of a matrix:

   \[\begin{split}\begin{array}{rclcl} \boldsymbol{A}=\boldsymbol{A}^+ & \Rightarrow & \det\boldsymbol{A}\ =\ \det\boldsymbol{A}^+ & & \\ & & \det\boldsymbol{A}\ =\ (\det\boldsymbol{A})^* & \Leftrightarrow & \det\boldsymbol{A}\in R\,. \end{array}\end{split}\]

3. If \(\,\boldsymbol{A}\in M_n(C)\ \) is a Hermitian matrix, then for every vector \(\ \boldsymbol{x}\in C^n\ \) an inner product \(\ \langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle\ \) is a real number:

   (5)\[\langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle\in R\,,\qquad \boldsymbol{x}\in C^n\,.\]

   Proof. \(\,\) Substitution \(\ \,\boldsymbol{A}^+=\boldsymbol{A},\ \ \boldsymbol{y}=\boldsymbol{x}\ \) in equation (3) leads to

   (6)\[\langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle\ =\ \langle\,\boldsymbol{A}\boldsymbol{x},\boldsymbol{x}\,\rangle\,,\qquad \boldsymbol{x}\in C^n\,.\]

   But since \(\ \,\langle\,\boldsymbol{A}\boldsymbol{x},\boldsymbol{x}\,\rangle= \langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle^*\,,\ \,\) we have \(\ \,\langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle= \langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle^*\,,\ \,\) and thus \(\ \,\langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle\,\in R\,.\)

   One can prove that the condition (5) is not only necessary, but also sufficient for a complex matrix \(\,\boldsymbol{A}\ \) to be Hermitian. This implies

   Corollary.

   If \(\ \boldsymbol{A}\in M_n(C)\,,\ \) then \(\qquad \boldsymbol{A}\ =\ \boldsymbol{A}^+\quad\Leftrightarrow\quad \langle\,\boldsymbol{x},\boldsymbol{A}\boldsymbol{x}\,\rangle\in R\,,\quad \boldsymbol{x}\in C^n\,.\)

4. For a Hermitian matrix \(\,\boldsymbol{A}\in M_n(C)\ \) the roots of characteristic polynomial \(\,w(\lambda)=\det\,(\boldsymbol{A}-\lambda\,\boldsymbol{I}_n)\ \) are real numbers.

   Proof.

   If \(\ \det\,(\boldsymbol{A}-\lambda\,\boldsymbol{I}_n)=0\,,\ \) then homogeneous linear problem with matrix \(\,\boldsymbol{A}-\lambda\,\boldsymbol{I}_n\ \) \(\\\) has nonzero solutions. \(\,\) Hence, there exists a nonzero vector \(\,\boldsymbol{x}\in C^n\ \,\) for which

   \begin{eqnarray*} (\boldsymbol{A}-\lambda\,\boldsymbol{I}_n)\ \boldsymbol{x} & \! = \! & \boldsymbol{0}\,, \\ \boldsymbol{A}\,\boldsymbol{x} & \! = \! & \lambda\,\boldsymbol{I}_n\,\boldsymbol{x}\,, \\ \boldsymbol{A}\,\boldsymbol{x} & \! = \! & \lambda\,\boldsymbol{x}\,, \quad\text{where}\quad\boldsymbol{x}\neq\boldsymbol{0}\,. \end{eqnarray*}

   If we substitute the last equality to the formula (6), we obtain

   \begin{eqnarray*} \langle\,\boldsymbol{x},\boldsymbol{A}\,\boldsymbol{x}\,\rangle & \! = \! & \langle\,\boldsymbol{A}\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \\ \langle\,\boldsymbol{x},\,\lambda\,\boldsymbol{x}\,\rangle & \! = \! & \langle\,\lambda\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \\ \lambda\ \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle & \! = \! & \lambda^*\;\langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \quad\text{where}\quad\langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle>0\,; \\ \lambda & \! = \! & \lambda^* \quad\ \ \Leftrightarrow\quad\ \ \,\lambda\in R\,. \end{eqnarray*}

Real Hermitian matrix is a symmetrix matrix: \(\,\) for \(\ \boldsymbol{A}\in M_n(R)\),

\[\boldsymbol{A}=\boldsymbol{A}^+\quad\Leftrightarrow\quad\boldsymbol{A}=\boldsymbol{A}^T\,.\]

Unitary Matrices

Definition.

Matrix \(\ \boldsymbol{B}\in M_n(C)\ \,\) is \(\,\) unitary \(\,\) if a product of Hermitian conjugate of a \(\\\) matrix \(\boldsymbol{B}\ \) and \(\,\) the matrix \(\boldsymbol{B}\ \) itself is an identity matrix:

(7)\[\boldsymbol{B}^+\boldsymbol{B}\,=\,\boldsymbol{I}_n\,.\]

\(\;\)

Example. \(\qquad\boldsymbol{B}\ =\ \displaystyle\frac{1}{\sqrt{2}}\ \left[\begin{array}{rr} 1 & i \\ i & 1 \end{array}\right]\,;\qquad \boldsymbol{B}^+\ =\ \displaystyle\frac{1}{\sqrt{2}} \left[\begin{array}{rr} 1 & -i \\ -i & 1 \end{array}\right]\,;\)

\[\begin{split}\boldsymbol{B}^+\boldsymbol{B}\ \ =\ \ \frac{1}{2}\ \left[\begin{array}{rr} 1 & -i \\ -i & 1 \end{array}\right]\ \left[\begin{array}{rr} 1 & i \\ i & 1 \end{array}\right]\ \ =\ \ \frac{1}{2}\ \left[\begin{array}{rr} 2 & 0 \\ 0 & 2 \end{array}\right]\ \ =\ \ \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right]\,.\end{split}\]

\(\;\)

Properties of unitary matrices. \(\\\)

0. The condition (7) means that \(\,\boldsymbol{B}^+=\boldsymbol{B}^{-1},\ \) which futrher implies that \(\,\boldsymbol{B}\boldsymbol{B}^+\,=\,\boldsymbol{I}_n\,.\ \) Hence, a unitary matrix \(\,\boldsymbol{B}\ \) satisfies identities

   \[\boldsymbol{B}^+\boldsymbol{B}\,=\,\boldsymbol{B}\boldsymbol{B}^+\,=\,\boldsymbol{I}_n\,.\]

1. The condition \(\ \boldsymbol{B}\boldsymbol{B}^+=\boldsymbol{I}_n\ \) may be written as \(\ (\boldsymbol{B}^+)^+\boldsymbol{B}^+=\boldsymbol{I}_n\,,\ \) which means that if \(\ \boldsymbol{B}\in M_n(C)\ \) is a unitary matrix, then so are the Hermitian conjugate \(\ \boldsymbol{B}^+\ \) and the inverse matrix \(\ \boldsymbol{B}^{-1}\,.\)

2. Let \(\ \boldsymbol{B}_1,\boldsymbol{B}_2\in M_n(C)\ \) be unitary matrices: \(\ \ \boldsymbol{B}_1^+\,\boldsymbol{B}_1=\boldsymbol{B}_2^+\,\boldsymbol{B}_2= \boldsymbol{I}_n\,.\ \) Then, by properties of Hermitian conjugation of matrices,

   \[(\boldsymbol{B}_1\boldsymbol{B}_2)^+(\boldsymbol{B}_1\boldsymbol{B}_2)\ =\ \boldsymbol{B}_2^+\,(\boldsymbol{B}_1^+\boldsymbol{B}_1)\,\boldsymbol{B}_2\ =\ \boldsymbol{B}_2^+\,\boldsymbol{I}_n\,\boldsymbol{B}_2\ =\ \boldsymbol{B}_2^+\,\boldsymbol{B}_2\ =\ \boldsymbol{I}_n\,.\]

   Hence, a product of unitary matrices is also a unitary matrix. \(\\\) In this way, because an identity matrix \(\ \boldsymbol{I}_n\ \) is unitary, we may write

   Corollary 1.

   A set of unitary matrices of size \(\,n\ \) together with matrix multiplication comprises a (nonabelian) group.

3. An inner product of the \(\,i\)-th and the \(\,j\)-th column of a unitary matrix \(\,\boldsymbol{B}\ \) is given by

   \[\langle\,\boldsymbol{b}_i,\boldsymbol{b}_j\rangle\ \,=\ \, \boldsymbol{b}_i^+\,\boldsymbol{b}_j\ \,=\ \, \left(\boldsymbol{B}^+\boldsymbol{B}\right)_{ij}\ \,=\ \, \left(\boldsymbol{I}_n\right)_{ij}\ \,=\ \,\delta_{ij}\,,\qquad i,j=1,2,\dots,n\,,\]

   because \(\,\boldsymbol{b}_i^+\ \) is the \(\,i\)-th row of a matrix \(\,\boldsymbol{B}^+,\ \ i=1,2,\dots,n.\)

   Taking into account the fact that matrix \(\,\boldsymbol{B}^+,\ \) whose columns are Hermitian conjugates of rows of matrix \(\,\boldsymbol{B},\ \) is also unitary, we may write

   Corollary 2.

   Matrix \(\ \boldsymbol{B}\in M_n(C)\ \) is unitary if and only if its columns \(\\\) (and also rows) \(\,\) comprise an orthonormal system in the space \(\,C^n.\)

4. Unitary matrix \(\,\boldsymbol{B}\in M_n(C)\ \) preserves an inner product in the space \(\,C^n:\)

   \[\langle\,\boldsymbol{B}\boldsymbol{x},\,\boldsymbol{B}\boldsymbol{y}\,\rangle\ \,=\ \, \langle\boldsymbol{x},\boldsymbol{y}\rangle\,,\qquad \boldsymbol{x},\boldsymbol{y}\in C^n\,.\]

   Indeed, by definition of an inner product in the space \(\,C^n,\ \) we have

   \[ \begin{align}\begin{aligned}\langle\,\boldsymbol{B}\boldsymbol{x},\,\boldsymbol{B}\boldsymbol{y}\,\rangle\ =\ (\boldsymbol{B}\boldsymbol{x})^+(\boldsymbol{B}\boldsymbol{y})\ =\ (\boldsymbol{x}^+\boldsymbol{B}^+)(\boldsymbol{B}\boldsymbol{y})\ =\\\ =\ \boldsymbol{x}^+(\boldsymbol{B}^+\boldsymbol{B})\ \boldsymbol{y}\ =\ \boldsymbol{x}^+\boldsymbol{I}_n\,\boldsymbol{y}\ =\ \boldsymbol{x}^+\boldsymbol{y}\ =\ \langle\boldsymbol{x},\boldsymbol{y}\rangle\,.\end{aligned}\end{align} \]

   In particular, if \(\,\boldsymbol{y}=\boldsymbol{x},\ \) we get an identity

   (8)\[\langle\,\boldsymbol{B}\boldsymbol{x},\,\boldsymbol{B}\boldsymbol{x}\,\rangle\ \,=\ \, \langle\boldsymbol{x},\boldsymbol{x}\rangle\,,\qquad \boldsymbol{x}\in C^n\,,\]

   which describes behaviour of the norm: \(\quad\|\,\boldsymbol{B}\boldsymbol{x}\,\|= \|\boldsymbol{x}\|\,,\ \ \boldsymbol{x}\in C^n\,.\)

   The last property allows to interpret multiplication (on the left hand side) of vector \(\,\boldsymbol{x}\in C^n\ \) by a unitary matrix \(\,\boldsymbol{B}\ \) as a generalised rotation of this vector.

5. Determinant of a unitary matrix \(\,\boldsymbol{B}\ \) is a complex number of modulus 1: \(\ \,|\det\boldsymbol{B}\,|=1\,.\)

   Indeed, applying determinant on both sides of the equality (7), we obtain

   \[\det\,(\boldsymbol{B}^+\boldsymbol{B})= \det\boldsymbol{B}^+\cdot\,\det\boldsymbol{B}= (\det\boldsymbol{B})^*\cdot\,\det\boldsymbol{B}= |\det\boldsymbol{B}\,|^2\quad=\quad \det\boldsymbol{I}_n=1\,.\]

6. For a unitarny matrix \(\,\boldsymbol{B}\in M_n(C)\ \) the roots of a characteristic polynomial \(\,w(\lambda)=\det\,(\boldsymbol{B}-\lambda\,\boldsymbol{I}_n)\ \) are complex numbers of modulus 1.

   Proof. \(\,\) If \(\ \det\,(\boldsymbol{B}-\lambda\,\boldsymbol{I}_n)=0\,,\ \) then homogeneous linear problem with matrix \(\,\boldsymbol{B}-\lambda\,\boldsymbol{I}_n\ \) \(\\\) has nonzero solutions: \(\,\) there exists a nonzero vector \(\,\boldsymbol{x}\in C^n\ \,\) for which

   \begin{eqnarray*} (\boldsymbol{B}-\lambda\,\boldsymbol{I}_n)\;\boldsymbol{x} & \! = \! & \boldsymbol{0}\,, \\ \boldsymbol{B}\,\boldsymbol{x} & \! = \! & \lambda\,\boldsymbol{I}_n\,\boldsymbol{x}\,, \\ \boldsymbol{B}\,\boldsymbol{x} & \! = \! & \lambda\,\boldsymbol{x}\,, \quad\text{where}\quad\boldsymbol{x}\neq\boldsymbol{0}\,. \end{eqnarray*}

   If we substitute the last equality to the formula (8), we obtain

   \begin{eqnarray*} \langle\,\boldsymbol{B}\boldsymbol{x},\,\boldsymbol{B}\boldsymbol{x}\,\rangle & \! = \! & \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \\ \langle\,\lambda\,\boldsymbol{x},\,\lambda\,\boldsymbol{x}\,\rangle & \! = \! & \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \\ \lambda^*\lambda\ \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle & \! = \! & \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \\ |\lambda|^2\ \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle & \! = \! & \langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle\,, \quad\text{where}\quad\langle\,\boldsymbol{x},\boldsymbol{x}\,\rangle>0\,; \\ |\lambda|^2 & \! = \! & 1 \quad\Rightarrow\quad|\lambda|=1\,. \end{eqnarray*}

Relation of a unitary matrix with a generalised rotation is also suggested by

Theorem 7.

Let \(\,V(C)\ \) be a unitary finite dimensional vector space with an orhonormal basis \(\,\mathcal{B}.\) \(\\\) Basis \(\,\mathcal{C}\ \) of this space is orthonormal if and only if a transition matrix \(\,\boldsymbol{S}\ \) from basis \(\,\mathcal{B}\ \) to \(\,\mathcal{C}\ \) is unitary.

Proof. \(\,\) Let \(\ \ \dim V=n\,,\ \ \mathcal{B}=(u_1,u_2,\dots,u_n)\,,\ \ \mathcal{C}=(w_1,w_2,\dots,w_n)\,,\ \ \boldsymbol{S}=[\,\sigma_{ij}\,]_{n\times n}\,.\)

By assumption, basis \(\,\mathcal{B}\ \) is orthonormal: \(\quad\langle u_i,u_j\rangle\,=\,\delta_{ij}\,,\quad i,j=1,2,\dots,n.\)

Definition of transition matrix implies the relations: \(\quad w_j\ =\ \displaystyle\sum_{i\,=\,1}^n\ \sigma_{ij}\,u_i\,,\quad j=1,2,\dots,n.\)

Consider an inner product of two vectors from basis \(\,\mathcal{C}\ \ (i,j=1,2,\dots,n):\)

\[\begin{split}\begin{array}{ccccc} \langle w_i,w_j\rangle & = & \left\langle\ \displaystyle\sum_{k\,=\,1}^n\ \sigma_{ki}\,u_k\,,\ \sum_{l\,=\,1}^n\ \sigma_{lj}\,u_l\right\rangle\ \,=\ \, \displaystyle\sum_{k,\,l\,=\,1}^n \sigma_{ki}^*\,\sigma_{lj}\,\langle u_k,u_l\rangle & = & \\ & = & \displaystyle\sum_{k,\,l\,=\,1}^n\ \sigma_{ki}^*\ \sigma_{lj}\ \delta_{kl}\ \ \,=\ \ \, \displaystyle\sum_{k\,=\,1}^n\ \sigma_{ki}^*\ \sigma_{kj}\ \ \,=\ \ \, \displaystyle\sum_{k\,=\,1}^n\ \sigma_{ik}^+\ \sigma_{kj} & = & \left(\,\boldsymbol{S}^+\boldsymbol{S}\,\right)_{ij}\ . \end{array}\end{split}\]

This implies in particular that

\[\langle w_i,w_j\rangle\ =\ \delta_{ij}\qquad\Leftrightarrow\qquad \left(\,\boldsymbol{S}^+\boldsymbol{S}\,\right)_{ij}=\delta_{ij}= \left(\,\boldsymbol{I}_n\right)_{ij}\,,\qquad i,j=1,2,\dots,n,\]

that is, basis \(\,\mathcal{C}\ \) is orthonormal if and only if \(\ \boldsymbol{S}^+\boldsymbol{S}=\boldsymbol{I}_n.\) \(\\\)

A real unitary matrix is an orthogonal matrix. Namely, for \(\ \boldsymbol{B}\in M_n(R):\)

\[\boldsymbol{B}^+\boldsymbol{B}=\boldsymbol{I}_n \quad\Leftrightarrow\quad \boldsymbol{B}^T\boldsymbol{B}=\boldsymbol{I}_n\,.\]